On two subgroups of U(n), useful for quantum computing

Alexis De Vos,Stijn De Baerdemacker

doi:10.1088/1742-6596/597/1/012030

Abstract

As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit ϕ(θ) and the 1-qubit NEGATOR circuit N(θ). Both are roots of the IDENTITY circuit. Indeed: both ϕ(0) and N(0) equal the 2 × 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2w); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2w). Here, ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n × n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n × n unitary diagonal matrices with first entry equal to 1. Any U(n) matrix can be decomposed into four parts: U = exp(iα) Z1XZ2, where both Z1 and Z2 are ZU(n) matrices and X is an XU(n) matrix. We give an algorithm to find the decomposition. For n = 2w it leads to a four-block synthesis of an arbitrary quantum computer.

Highlights

The unitary group U(n) is important for quantum computing, because all quantum circuits acting on w qubits can be represented by a member of the unitary group U(2w)
Within U(2), the I matrix has a lot of square roots: four diagonal matrices
N (θ), i.e. a PHASOR acting on the first qubit and a NEGATOR acting on the second qubit, respectively

Summary

Introduction

The unitary group U(n) is important for quantum computing, because all quantum circuits acting on w qubits can be represented by a member of the unitary group U(2w). 2. Decomposition of an arbitrary unitary matrix Two-qubit circuits are represented by matrices from U(4). N (θ) , i.e. a PHASOR acting on the first qubit and a NEGATOR acting on the second qubit, respectively These circuits are represented by the 4 × 4 unitary matrices. Because the multiplication of two square matrices with all line sums equal to 1 automatically yields a third square matrix with all line sums equal to 1, we can demonstrate that an arbitrary quantum circuit like doi:10.1088/1742-6596/597/1/012030 consisting merely of uncontrolled NEGATORs and controlled NEGATORs is represented by a 2w × 2w unitary matrix with all line sums equal to 1. The n × n unitary diagonal matrices with upper-left entry equal to 1 form a group ZU(n), subgroup of U(n).

H X2 H X H X1 H

Conclusion